mathematics//linear algebra//eigenvalue//non-normal matrix

A non-normal matrix is a square matrix whose eigenvectors are not perpendicular to one another (formally, one that does not commute with its transpose), and it matters to engineers because such a system can amplify a disturbance many times over before its stable eigenvalues bring it back down. Eigenvalues tell the long-term fate of \(x_{k+1}=Ax_k\), never the path to it. Take

A non-normal matrix is a square matrix whose eigenvectors are not perpendicular to one another (formally, one that does not commute with its transpose), and it matters to engineers because such a system can amplify a disturbance many times over before its stable eigenvalues bring it back down. Eigenvalues tell the long-term fate of xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​, never the path to it. Take

A=[0.9500.9].A=\begin{bmatrix}0.9 & 5\\ 0 & 0.9\end{bmatrix}.A=[0.90​50.9​].

Both eigenvalues are 0.9, stable without discussion, and yet the norm of AkA^kAk climbs to about 19 around step 10 before it decays. The reason is visible in the matrix: the first state decays, but the second keeps feeding it through the 5, and the feeding outpaces the decay for a while. Its two eigenvectors are almost parallel, so a modest vector has to be written as a difference of two huge modal components, and when those decay at the same rate their difference first grows.

Stable eigenvalues guarantee eventual decay and promise nothing about the way there.

A transient of 19 times saturates an actuator and wakes up nonlinearities long before asymptotic stability offers any comfort. To see the transient, look at the singular values of AkA^kAk, or of eAte^{At}eAt in continuous time.

The pattern is typical of subsystems connected in series, where each one feeds the next and nothing feeds back: a chain of tanks, a cascade of filters, a convoy of vehicles each following the one ahead. In continuous time the matrix with eigenvalues −1-1−1 and −2-2−2 and a coupling of 100 from the second state into the first amplifies a disturbance on the second state by about 25 times before it dies out.

Symmetric matrices are normal, and so are rotations; for them eigenvalues tell the whole story, and the norm of AkA^kAk is exactly ρ(A)k\rho(A)^kρ(A)k. Covariances, graph Laplacians and the Hessian of a loss are safe; dynamics matrices of real plants, closed loops A−BKA-BKA−BK and the Jacobians of recurrent networks usually are not.

The spectral radius still decides the end, which is why a simulation run long enough looks fine. The failure shows up in the first seconds after a step or a gust, when a state the designer considered quiet hits a limit and the saturation makes the loop nonlinear.

Near-parallel eigenvectors also make the eigenvectors themselves fragile: a tiny perturbation of the matrix rotates them far, so they should not be read as physical shapes.